Remark 3 A more general definition is as follows:
Let \(X\)  be a smooth projective variety and and \(Z_1,Z_2\subset X\)  be two irreducible subvariety of \(X\) . We say that \(Z_1\)  and \(Z_2\)  are flat algebraically equivalent if we have have holomorphic maps \(f: Z\rightarrow (\mathbb {C}^m,0)\)  and \(g: Z\rightarrow X\)  such that \(f\)  is flat and \(g\)  restricted to the fibers of \(f\)  is an injection.
Duco van Straten kindly reminded me [January 14, 2019]
that
“the requirement of flatness of a
cycle is not a reasonable condition at
all. Flatness is just too special.
There is the notion of families of
cycles used by Barlet in analytic
and by Kollar in algebraic geometry[see [1]}, page 45-46 and [2]}].
It is very ugly and algebraically hard
to use, but geometrically reasonable."
For our purpose we need Proposition REF  which is very specific to rational curves. One might expect that if two irreducible algebraic cycle \(Z_1,Z_2\subset X\)  are flat algebraically equivalent then they are homologically equivalent.
It is not known to the author whether this is true for rational curves.

